Department of Electronic Engineering ELE2EMI Electronic Measurements & Instrumentation

2 Bridge Circuits

Chapter references:

• Berlin & Getz, chapter 9.

• Carr, chapter 4.

Definition: Bridges are electrical circuits for performing null measurements on resistances in DC () and general impedances in AC ().

2.1 DC Bridges

2.1.1

The DC bridge shown below is known as the Wheatstone bridge. It is used to obtain a precise measurement (accurate to about 0.1%) of a RX . (Usually it will be represented in a diamond shape, but it’s easier for me to draw the vertically or horizontally, and i think the circuit is slightly easier to understand this way.)

R Meter R + 1 + − 3 VS V 2 V X − R 2 R X

This bridge consists of two resistor branches: on the left are two precision resistors R1 and R2. On the right are a precisely calibrated variable resistor R3 and an unknown (or imprecisely known) resistor RX . If the current flowing between the mid-points of the branches happens to be zero, then the branches will function as accurate dividers, in which case the voltage across R2 will be

R2 V2 = VS R1 + R2 and this will equal the voltage across the unknown resistor RX which will be

1 RX VX = RX + R3

We obtain this equality by carefully adjusting R3. In the diagram a sensitive current meter, a , is used to detect equality of the . In order not to risk damage to the meter, it’s advisable to obtain approximate balance before inserting the meter: we can estimate a safe starting value for R3 by first measuring RX approximately using a less precise instrument such as digital .

A zero-centre galvanometer is the most suitable type, as it can measure small changes in either direction as we adjust R3. If the meter has no significant zero error, then when it reads zero current the ratios of the resistors in both dividers must agree, leading to the null condition formula:

R1RX = R2R3

As Carr section 4-5 explains, the current through the galvanometer is proportional to any small change in the value of one of the resistors. This enables the indirect measurement of quantities that affect the resistivity of certain materials. Such are described in more detail in a subsequent lecture.

2.2 AC Bridges

The general alternating current (AC) bridge is used to measure inductors and capacitors. It has the same form as a Wheatstone bridge, but the voltage source is now AC, general impedances replace the four resis- tances, and the null detector is no longer a galvanometer but an instrument suited to AC measurement.

Z Meter Z 1 + − 3 vs v2 vx

Z 2 Z X

The form of the null condition remains the same:

Z1ZX = Z2Z3 but these are now complex quantities, so this leads to two balance equations. The resistive component is usually balanced in the same way as in the Wheatstone bridge using a precisely calibrated variable resistance, whereas the reactive component is balanced by a similar impedance (capac- itance for , or for inductance) in an adjacent arm of the bridge, or by an opposite impedance (inductance balancing capacitance) in an opposite arm. Variable inductors and capacitors are not so easy to make as variable resistors, so to obtain the effect of a variable precision reactance, most circuits use a fixed precision capacitor with a variable resistance (either in series or in parallel with it).

2 2.2.1 AC Null Detectors

For precise measurement of a null condition in the presence of time-varying voltages and currents, the galvanometer must be replaced by instruments such as AC or . (As many of you will recall from Electronics 1, this school makes much use of oscilloscopes for AC measurements.) Differential amplifiers (in addition to those already contained in the ) can be employed to in- crease the sensitivity of the measurement.

2.2.2 AC Isolation

A difficulty you may already have encountered with AC signals is that they are sensitive to the presence of objects that are not in obvious electrical contact with them, for example a human standing perhaps 0.3 metre away. This is because the changing electric fields in AC are not confined to conductive wires, but extend through space. In effect, a nearby human contributes capacitance to the AC circuit. The human operator needs to be near the controls and the meter, but perhaps these can be isolated from the AC bridge. Carr (page 97 and Figure 4-5 on page 98) suggests a transformer or a differential amplifier to bring the signal to a coaxial cable and then to the meter.

2.2.3 Variations

Reference: Berlin & Getz, section 9-4. Capacitors are not ideal , but contain a resistive component; likewise for resistors. A typical capacitor model is a capacitance CP in parallel with a leakage resistance RP . Likewise a realistic inductor is commonly modelled as an inductor LS in series with a resistance RS. However, the opposite models can also be used, as shown below.

R S C P R P

C S

R S R P L P

L S

It’s important to recognise that the values of the components in the two types of model are not equal, though they are related. For the capacitor,

D2 RS = RP 1+ D2

2 CS = CP (1 + D )

The dissipation factor

3 XP 1 RS D = = = = ωCSRS RP ωRP CP XS is proportional to the power dissipated (lost as heat) through the capacitor each cycle of the AC signal. A good (nearly ideal) capacitor has a low dissipation factor. For the inductor, the relations are:

2 RP = RS(1 + Q ) 1 LP = LS(1 + ) Q2 where the quality factor Q is

XS ωLS RP RP Q = = = = RS RS XP ωLP A good inductor has a high quality factor. Note that the formulas for D and Q are reciprocal in form: this is a consequence of how engineers chose to define them. For both capacitors and inductors, the ideal is that RS should be small and RP should be large. Incidentally, in frequency dependent circuits, the quality factor Q describes how well-tuned the circuit is. This is useful to know when designing, for example, radio receivers. A circuit with a high Q can more easily filter out unwanted radio stations while giving good reception of the desired station. Since a real inductor has a nonzero resistance, it is a simple tuned circuit in its own right (as are real capacitors for the same reason). It’s important to note that the series and parallel models have limited ranges of applicability, and the model LS and LP and capacitances CS and CP may themselves vary with frequency in ways that can only be determined experimentally. We shall now describe a few of the more common AC bridges.

2.2.4 Schering's Bridge

Schering’s bridge is used to measure capacitance. It may also be used to measure the capacitor’s dissipation factor, and the resistances of cables and equipment. (Note that ‘Schering’ and ’capacitor’ both contain the letter “C”.)

C 1 R 1 R 3

vs

C X C unknown 2 capacitor R X

Its balance equations are independent of frequency:

C1 RX = R1 C2 R1 CX = C2 R3

4 2.2.5 Maxwell's Bridge

R 1 C 1 R 3 vs

L X R unknown 2 inductor R X

In Maxwell’s bridge, resistors R2 and R3 remain as in Wheatstone’s bridge, but a variable resistor R1 is placed in parallel with a standard (precisely known) capacitor C1. A series model is used for the unknown inductor. Note that what amounts to an appreciable resistance depends heavily on the frequency range. At low frequencies, resistance dominates inductance, not capacitance, but at high frequencies it dominates capaci- tance, not inductance. In Maxwell’s bridge, the resistive null condition is identical to that in the Wheatstone bridge:

R2R3 RX = R1 and the reactive null condition relates LX to C1:

LX = C1R2R3

Conveniently, the frequency does not appear in either balance equation, so one does not need to make a frequency measurement to measure the inductance.

The can be used to measure the Q of the unknown inductor (LX in series with RX ) provided that the frequency is determined:

Q = ωR1C1

Maxwell’s bridge is suited to measuring low-Q inductors (Q between 1 and 10). It helps to remember that both ‘Maxwell’ and ‘low’ contain the letter ‘L’.

5 2.2.6 Hay's Bridge

R 1

C 1 R 3 vs

L X R unknown 2 inductor R X

In Hay’s bridge, R1 is in series with C1. Hay’s bridge is less convenient for measuring inductances, as its balance equations

C1R2R3 LX = 2 1+(ωR1C1)

2 (ωC1) R1R2R3 RX = 2 1+(ωR1C1) would usually require us to make frequency measurements. However, for inductors with high Q, this is less significant. Hay’s bridge is suited to measuring high-Q inductors (Q above 10). Think ‘H’ for ‘Hay’ and ‘high’), and therefore complements Maxwell’s bridge.

2.2.7 Owen's Bridge

C 1 R 3

vs R 2 L X unknown C inductor 2 R X

The balance equations for Owen’s bridge are frequency independent:

6 LX = C1R2R3

C1 RX = R2 C2 The Owen bridge is suited to measuring a wide range of inductances. (There’s a ‘W’ in both ‘Owen’ and ‘wide’!) Geoffrey Tobin 2007 July 25

7